2003 BMW 325 stalls when accelerating

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My car will stall out when accelerating, and I have to pull over, restart the car, and keep going. When i try to accelerate it will lose all its power, and then after I stop it will start to bouncing between higher and lower rpm’s. Restarting fixing for a little bit bout won’t make it very far.

Is the Check Engine Light lit-up?
Or–even worse–is the CEL flashing/blinking?

If you are lucky, your fuel pump is failing. That is a common problem on the e46. Both mine are on their second fuel pumps. Actually, if you are REALLY lucky, it is just the fuel filter, but I would not count on that. Either way, these are repairs that you can do yourself for not a lot of money if you know how to be careful around gasoline fumes.

If your car consumes a lot of oil (another common problem on e46s with over 100k miles) then you may have plugged catalytic converters. I went through that drill with one of mine. Catalytic converters are 2 x $1000 ea for California legal converters, and you have to safely support the engine and remove the right engine mount to replace them. If you cannot easily feel the individual bursts of exhaust pressure at the tail pipe when the car is idling, that is a bad sign. If the computer sets a code for “MAF signal implausible” that is also a bad sign. It means that the computer cannot understand why not enough air is coming in for the RPM and throttle opening, and the reason is because the exhaust cannot get out.